Revisiting a connection between arithmetic and geometry

I saw a really great thread on twitter this week and wanted to share some of the ideas with the boys for our Family Math project today:

My observation that you can sum the first n cubes to get 1^3 + … + n^3 = [(n*(n+1)]^2 by counting rectangles in an nxn square is of course well-known (as is everything simpler and clever).

Here's one paper:https://t.co/hSTcJcXm6b

— Christopher D. Long (@octonion) March 21, 2019

I like this proof of the square sum, shared by @jeremyjkun : https://t.co/JraIdYO0eJ

— Colin Beveridge (@icecolbeveridge) March 21, 2019

We started off looking at the sum 1 + 2 + 3 + . . . .

Next we looked at the sum of squares and searched for a geometric connection:

Now I showed them the fantastic way of looking at the sum of squares in the Jeremy Kun blog post. This method is a terrific way to share an advanced idea in math with kids – it is totally accessible to them and gives them a chance to talk through a fairly complicated idea:

Finally, I showed how the ideas we were just talking about extend to some of the basic ideas is calculus. It was neat to hear my younger son talking through the ideas here, too:

Definitely a neat morning – it is always amazing to see the connections between arithmetic and geometry.